Harmonic complex structures and special hermitian metrics on products of Sasakian manifolds

Abstract

It is well known that the product of two Sasakian manifolds carries a 2-parameter family of Hermitian structures (J_(a,b), g_(a,b)). We show in this article that the complex structure J_(a,b) is harmonic with respect to g_(a,b) , i.e. it is a critical point of the Dirichlet energy functional. Furthermore, we also determine when these Hermitian structures are locally conformally Kähler, balanced, strong Kähler with torsion, Gauduchon or k-Gauduchon (k ≥ 2). Finally, we study the Bismut connection associated to (J_(a,b), g_(a,b)) and we provide formulas for the Bismut-Ricci tensor Ric^B and the Bismut-Ricci form ρ^B . We show that these tensors vanish if and only if each Sasakian factor is η-Einstein with appropriate constants and we also exhibit some examples fulfilling these conditions, thus providing new examples of Calabi-Yau with torsion manifolds.